delta09 wrote:
x is the sum of y consecutive integers. w is the sum of z consecutive integers. If y = 2z, and y and z are both positive integers, then each of the following could be true EXCEPT
A. x = w
B. x > w
C. x/y is an integer
D. w/z is an integer
E. x/z is an integer
For these kind of questions, we need to parse through the options to find an option which can NEVER be true. There are three scenarios possible with any option:
1. We can easily find out that this is possible. Then, we can eliminate this option and move forward.
2. We can easily find out that the option is never true. Thus, we can stop here because this is the answer.
3. Nothing can be easily inferred. We need to plug in values and do some reasoning to figure out. These options can be skipped for the moment to come back later, if we don't find an answer among other options. We want to SAVE our time.
Here is my attempt at this:
A. x = w - just looking at it, i can't say if this is possible or not. To find so will take time.I skip it to come back later
B. x > w - Easily possible is x is the sum of first 2 integers and w is the sum of first 1 integer
C. x/y is an integer - come back later
D. w/z is an integer - easily possible if z=1, w=1
E. x/z is an integer[/quote] - easily possible if z=1, x=3
now, we are left with A and C options. I begin with option C.
x=sum of consecutive y (or 2z) integers = n + (n+1) + (n+2) +....+ (n+2z-1) = 2zn + (1+2+3+...+2z-1) = 2zn + 2z*(2z-1)/2 (using formula for sum of first n natural numbers)
=> x/y or x/2z = n + (2z-1)/2
we know n is an integer. The other component can never be an integer, since the numerator is always odd and denominator is 2. So, x/y cannot be an integer.
Thus, C is the answer.
Cheers,
CJ
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